What is Symplectic Geometry? (2016) [pdf]

[ajkjk]

This really doesn't make it clear what symplectic geometry... is, or why I should care about it. I have eventually figured out an answer that was satisfactory to me, after much frustration: it is math on a manifold that has a concept of paired-off coordinates, like (x,v) in mechanics. Typically this is interesting because it is an alternate characterization of the mathematics of a space where the relevant quantities are a variable and its derivative.

(In classic mechanics, particularly, there is an unusual symmetry to position and velocity, such that the laws of mechanics look roughly a rotation x -> v, v -> -x, which is why this works so well.)

[danharaj]

> (In classic mechanics, particularly, there is an unusual symmetry to position and velocity, such that the laws of mechanics look roughly a rotation x -> v, v -> -x, which is why this works so well.)

In fact, every symplectic manifold locally looks like this (Darboux's theorem).

[turnersr]

What does "paired-off coordinates, like (x,v) in mechanics" mean? What is x and v here?

Thanks!

[Ceezy]

True, the main reason is to understand differential equations. The first vector is supposed to be the speed and the second the acceleration. And the area represented by the dqdp is invariant through time in a conservative system.

[pjbk]

Correct, and it also appears in Electromagnetism and Quantum Physics (in several ways), and in Lie theory that is useful for rotations and differential applications.

[throwlaplace]

Symplectic geometry is a collection of facts having to do with symplectic manifolds. Just like euclidean geometry is a collection of facts having to do euclidean manifolds.

That's the way in which terms like "____ geometry" are defined and understood.

[Ragib_Zaman]

Quite interestingly, Symplectic Geometry is currently under review/investigation for some of the foundational papers in the field having serious gaps and outright errors after closer inspection. These concerns were always spoken of in hush hush tones and only in recent times have people stated their concerns publically. Some of the original authors refuse to retract their papers despite being assured their academic positions (which realistically, came through the reputation built up by these papers) are secure. Here's a quanta article about this fiasco: https://www.quantamagazine.org/the-fight-to-fix-symplectic-g...

[sidek]

As a result, lots of recent work is being done in the algebraic setting (rather than analytic), where the foundations are on much firmer footing.

Being algebraic symplectic is a much stronger condition than analytic symplectic, but is still interesting enough (and, for geometry related to linear algebra problems, as is often relevant in CS, is not a very strong restriction at all.)

[messe]

That said, it's worth noting that the parts that are of interest to someone new to Symplectic Geometry, tend to be those related to Hamiltonian Mechanics or similar, and are likely not under review. Most of Symplectic Geometry is "probably" (I'm using quotes and italics to hedge my bets) "fine"-ish.

[pjbk]

For those into physics, I wholeheartedly recommend Marsden and Ratiu's book, "Introduction to Mechanics and Symmetry", which deals mainly with the different formulations of physics applied to symplectic and associated geometries.

[tobmlt]

Thanks for the recommendation! I didn't know this one. I try and lap up everything I can by Marsden, (though more often through the lens of applied researchers: e.g. Desbrun, Hirani, Crane, and others -- much involving computer graphics and/or discrete differential geometry applied to physical simulation. In short, I better say that I am not familiar with the scope of Marsden's work. I am sure much of it is beyond me, but gosh darned it, the exterior calculus is beautiful and these guys write brilliantly readable stuff for an engineer.

Even as a hydrodynamics software guy, I found the computer graphics research community to be the easiest entry-point for, especially, the topology and modern differential geometry. It's especially nice when they do a simulation paper with a high end geometric/analytic approach.

This might be a good place to go in order to have a start at, say, Arnold's ``topological methods in hydrodynamics'' or anything TQFT-esque.

[evanb]

Importantly, the Hamiltonian formulation of classical mechanics has symplectic form, with the conjugate variables (position, momentum) making up the dimensions.

[killjoywashere]

@agronaut: most interesting to me: why did you post this? What were you working on that led you to this?

[madrafi]

This was quite a treat !

[tomrod]

This seems to have crossover with topological data analysis.

[akimball]

In particular, for geometrizing semantics. Montague grammar is a tarpit, and pragmatic utility of inference on distributed representations has been abundantly demonstrated in the past decade. Symplectic structure is one of a small class of structures which capture and relate essential features of natural semantics in a metric (read, tractable) representation. This offers a tantalizing prospect for bridging the gap between computation and cognition.